Extremal approximations in the bandlimit and the Rayleigh criterion for super-resolution

Maxime Ferreira Da Costa (L2S, Centralesupelec)
April 26, 2024 — 11:00 — "new L2S location (IBM building), 3rd floor room" (and Teams)

Abstract

Of particular interest in imaging sciences and telecommunications is the super-resolution problem, which consists of recovering a stream of spikes (point sources) from the noisy observation of a few numbers of its first trigonometric moments weighted by the ones of the point-spread function (PSF). The empirical feasibility of this problem has been known since the work of Rayleigh on diffraction, which is essentially driven by the separation between the spikes to recover. We present a novel statistical framework based on the spectrum of the Fisher information matrix (FIM) to quantify the stability limit of super-resolution as a function of the PSF. In the regime where the minimal separation is inversely proportional to the number of acquired moments, we show the existence of a separation constant above which the minimal eigenvalue of the FIM is not asymptotically vanishing—defining a statistical resolution limit. Notably, a relationship between the total variation of the autocorrelation function of the PSF and its association resolution limit is highlighted.

Bio

Maxime Ferreira Da Costa received the M.Sc. degree in electrical engineering from Imperial College London, U.K., in 2012, the Diplôme d’Ingénieur degree from CentraleSupélec, Université Paris-Saclay, France, in 2012, and the Ph.D. degree in electrical engineering from Imperial College London in 2018. Since 2022, he has been an Associate Professor with CentraleSupélec, Université Paris–Saclay, France, working with the Laboratory of Signals and Systems. Before that, he was a Research Associate with the Department of Electrical and Computer Engineering, University of Southern California, from 2020 to 2022 and the Department of Electrical and Computer Engineering, Carnegie Mellon University, Pennsylvania, from 2018 to 2020. His research focuses on the mathematical foundations of data science and signal processing, and applied harmonic analysis. In 2018, he was shortlisted among the finalists for the Jack Keil Wolf Student Paper Award at the IEEE International Symposium on Information Theory.

References

[1] M. Ferreira Da Costa, “The condition number of weighted non-harmonic Fourier matrices with applications to super-resolution”, hal-04261330, 2023, preprint.

[2] M. Ferreira Da Costa, Y. Chi, “Local Geometry of Spike Deconvolution from Low-Pass Measurements,” IEEE Journal on Selected Areas in Information Theory, vol. 4, pp. 1-15, 2023.

[3] M. Ferreira Da Costa, “Second-Order Beurling Approximations and Super-Resolution from Bandlimited Functions,” Sampling Theory and Applications Conference (SampTA 2023), July 2023.